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Stability of reverse isoperimetric inequalities in the plane: area, Cheeger, and inradius

Kostiantyn Drach, Kateryna Tatarko

TL;DR

The paper proves sharp stability results for reverse-type isoperimetric inequalities in the plane under $\lambda$-convexity. By exploiting inner parallel geometry and reductions to $\lambda$-convex lenses, it establishes: (i) stability of the lens isoperimetric quotient $\mathcal{I}(K)$ with $d_{\mathcal{H}}(K,L) \le C\varepsilon^{1/4}$, (ii) a reverse Cheeger inequality with a corresponding stability $d_{\mathcal{H}}(K,L) \le C\varepsilon^{1/4}$, and (iii) stability for the reverse inradius quotient with $d_{\mathcal{H}}(K,L) \le C\sqrt{\varepsilon}$, all for $\lambda$-convex bodies in $\mathbb{R}^2$. Central to the approach are the inner parallel framework, Blaschke's rolling theorem, and a careful lens-based comparison that reduces general $\lambda$-convex bodies to extremal lens configurations. The results provide the first quantitative stability statements for reverse-type inequalities in the plane and suggest pathways to higher dimensions.

Abstract

In this paper, we present sharp stability results for various reverse isoperimetric problems in $\mathbb R^2$. Specifically, we prove the stability of the reverse isoperimetric inequality for $λ$-convex bodies -- convex bodies with the property that each of their boundary points $p$ supports a ball of radius $1/λ$ so that the body lies inside the ball in a neighborhood of $p$. For convex bodies with smooth boundaries, $λ$-convexity is equivalent to having the curvature of the boundary bounded below by $λ> 0$. Additionally, within this class of convex bodies, we establish stability for the reverse inradius inequality and the reverse Cheeger inequality. Even without its stability version, the sharp reverse Cheeger inequality is new in dimension $2$.

Stability of reverse isoperimetric inequalities in the plane: area, Cheeger, and inradius

TL;DR

The paper proves sharp stability results for reverse-type isoperimetric inequalities in the plane under -convexity. By exploiting inner parallel geometry and reductions to -convex lenses, it establishes: (i) stability of the lens isoperimetric quotient with , (ii) a reverse Cheeger inequality with a corresponding stability , and (iii) stability for the reverse inradius quotient with , all for -convex bodies in . Central to the approach are the inner parallel framework, Blaschke's rolling theorem, and a careful lens-based comparison that reduces general -convex bodies to extremal lens configurations. The results provide the first quantitative stability statements for reverse-type inequalities in the plane and suggest pathways to higher dimensions.

Abstract

In this paper, we present sharp stability results for various reverse isoperimetric problems in . Specifically, we prove the stability of the reverse isoperimetric inequality for -convex bodies -- convex bodies with the property that each of their boundary points supports a ball of radius so that the body lies inside the ball in a neighborhood of . For convex bodies with smooth boundaries, -convexity is equivalent to having the curvature of the boundary bounded below by . Additionally, within this class of convex bodies, we establish stability for the reverse inradius inequality and the reverse Cheeger inequality. Even without its stability version, the sharp reverse Cheeger inequality is new in dimension .
Paper Structure (13 sections, 9 theorems, 112 equations, 2 figures)

This paper contains 13 sections, 9 theorems, 112 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof of Theorem \ref{['Thm:A']} assuming Theorem \ref{['Thm:B']}
  4. Inner parallel bodies in $\mathbb{R}^n$.
  5. Proof of Theorem \ref{['Thm:A']}
  6. On the optimality of the exponent of $\varepsilon$
  7. Proof of Theorem \ref{['Thm:C']} assuming Theorem \ref{['Thm:B']}
  8. Proof of the reverse Cheeger inequality
  9. Proof of stability in the reverse Cheeger inequality
  10. Proof of Theorem \ref{['Thm:B']}
  11. The basic construction and preliminaries
  12. Conditional proof of Theorem \ref{['Thm:B']}
  13. Proof of Theorem \ref{['Thm:B']} without assuming \ref{['Eq:Touching']}

Key Result

Theorem A

For every $\lambda > 0$ there exist $\varepsilon_0 \in (0,1)$ and $C \geqslant 1$ such that for every positive $\varepsilon \leqslant \varepsilon_0$ the following holds: If $K \subset \mathbb{R}^2$ is a $\lambda$-convex body such that then there exists a $\lambda$-convex lens $L \subset \mathbb{R}^2$ such that where $d_{\mathcal{H}}$ is the Hausdorff distance.

Figures (2)

  • Figure 1: The definition of a $\lambda$-convex body
  • Figure 2: Graphs of $|\partial K_t|$ and $|\partial L_t|$ in the proof of Theorem \ref{['Thm:A']}

Theorems & Definitions (13)

  • Theorem A: Stability in the reverse isoperimetric inequality in $\mathbb{R}^2$
  • Corollary 1.1
  • Theorem B: Reverse Cheeger inequality and its stability
  • Remark
  • Theorem C: Stability in reverse inradius inequality in $\mathbb{R}^2$
  • Proposition 2.1
  • proof
  • Theorem 3.1: KLR
  • Lemma 4.1: Hausdorff distance from pieces
  • Lemma 4.2: Proximity of lenses
  • ...and 3 more